US8972164B2 - Collaborative robot manifold tracker - Google Patents
Collaborative robot manifold tracker Download PDFInfo
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- US8972164B2 US8972164B2 US13/868,796 US201313868796A US8972164B2 US 8972164 B2 US8972164 B2 US 8972164B2 US 201313868796 A US201313868796 A US 201313868796A US 8972164 B2 US8972164 B2 US 8972164B2
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01F—MEASURING VOLUME, VOLUME FLOW, MASS FLOW OR LIQUID LEVEL; METERING BY VOLUME
- G01F1/00—Measuring the volume flow or mass flow of fluid or fluent solid material wherein the fluid passes through a meter in a continuous flow
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01P—MEASURING LINEAR OR ANGULAR SPEED, ACCELERATION, DECELERATION, OR SHOCK; INDICATING PRESENCE, ABSENCE, OR DIRECTION, OF MOVEMENT
- G01P5/00—Measuring speed of fluids, e.g. of air stream; Measuring speed of bodies relative to fluids, e.g. of ship, of aircraft
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- G—PHYSICS
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- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
Definitions
- the invention is directed to tracking coherent structures and manifolds on flows in a designated fluid medium, and in particular to deploying mobile autonomous underwater flow sensors, e.g. each tethered to a watersurface craft, to track stable/unstable manifolds of general 2D conservative flows through local sensing alone.
- LCS Lagrangian coherent structures
- LCS are analogous to ridges defined by local maximum instability, and can be quantified by local measures of Finite-Time Lyapunov Exponents (FTLE) (S. C. Shadden, F. Lekien, and J. Marsden, “Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows,” Physica D: Nonlinear Phenomena, vol. 212, no. 3-4, pp. 271-304 (2005)). Recently, LCS have been shown to coincide with optimal trajectories in the ocean which minimize the energy and the time needed to traverse from one point to another (see, e.g., T. Inane, S.
- FTLE Finite-Time Lyapunov Exponents
- a collaborative control method for tracking Lagrangian coherent structures (LCSs) and manifolds on flows employs at least three autonomous sensors each equipped with a local flow sensor for sensing flow in a designated fluid medium, e.g. water or air.
- a first flow sensor is a tracking sensor while the other sensors are herding sensors for controlling and determining the actions of the tracking sensor.
- the tracking sensor is positioned with respect to the herding sensors in the fluid medium such that the herding sensors maintain a straddle formation across a boundary; obtaining a local fluid flow velocity measurement from each sensor.
- a global fluid flow coherent structure is predicted based on the local flow velocity measurements.
- mobile autonomous underwater flow sensors may be deployed with each tethered to a watersurface craft.
- the invention advantageously uses cooperative robots to find coherent structures without requiring a global picture of the ocean dynamics, and enables a team of robots to track the stable/unstable manifolds of general 2D conservative flows through local sensing alone.
- the invention provides tracking strategies for mapping LCS in the ocean using AUVs, using nonlinear dynamical and chaotic system analysis techniques to create a tracking strategy for a team of robots.
- the cooperative control strategy leverages the spatio-temporal sensing capabilities of a team of networked robots to track the boundaries separating the regions in phase space that support distinct dynamical behavior. Additionally, boundary tracking relies solely on local measurements of the velocity field.
- the method of the invention may be generally applied to any conservative flow.
- FIG. 1 shows three robots tracking the stable structure B S in a conservative vector field according to the invention
- FIGS. 2A-B show trajectories of 3 robots tracking a sinusoidal boundary ( FIG. 2A ) and a star-shaped boundary ( FIG. 2B ) according to the invention
- FIGS. 3A-B show trajectories of a 3 robot team tracking a star shape ( FIG. 3A ) and a snapshot of the multi-robot experiment ( FIG. 3B ) according to the invention
- FIG. 4 shows a phase portrait of a time-independent double-gyre model according to the invention.
- FIGS. 5A-5H show snapshots of the trajectories of the team of 3 robots tracking Lagrangian coherent structures according to the invention.
- B S and B U denote the stable and unstable manifolds of Eq. (2).
- B S and B U are the separating boundaries between regions in phase space with distinct dynamics.
- B * are simply one-dimensional curves where * denotes either stable (S) or unstable (U) boundaries. For a small region centered about a point on B * , the system is unstable in one dimension.
- the objective is to develop a collaborative strategy to enable a team of robots to track B * in general 2D planar conservative flow fields through local sampling of the velocity field. While the focus is on the development of a tracking strategy for B S , the method can be easily extended to track B U since B U are simply stable manifolds of Eq. (2) for t ⁇ 0.
- the method of the invention originates from the Proper Interior Maximum (PIM) Triple Procedure, H. E. Nusse and J. A. Yorke, “A procedure for finding numerical trajectories on chaotic saddles,” Physica D Nonlinear Phenomena, vol. 36, pp. 137-156, 1989 (hereinafter “Nusse et al.”)—a numerical technique designed to find stationary trajectories in chaotic regions with no attractors. While the original procedure was developed for chaotic dynamical systems, the approach can be employed to reveal the stable set of a saddle point of a general nonlinear dynamical system. The procedure consists of iteratively finding an appropriate PIM Triple on a saddle straddling line segment and propagating the triple forward in time.
- PIM Proper Interior Maximum
- J be a line segment that crosses the stable set B S in D, i.e., the endpoints of the J are on opposite sides of B S .
- ⁇ x L ,x C ,x R ⁇ denote a set of three points in J such that x C denotes the interior point.
- ⁇ x L ,x C ,x R ⁇ is an Interior Maximum triple if T E (x C )>max ⁇ T E (x L ),T E (x R ) ⁇ .
- ⁇ x L ,x C ,x R ⁇ is a Proper Interior Maximum (PIM) triple if it is an interior maximum triple and the interval [x L , x R ] in J is a proper subset off.
- PIM Proper Interior Maximum
- any PIM triple can be obtained iteratively starting with an initial saddle straddle line segment J 0 .
- T E (q i ) For every point q i , determine T E (q i ) by propagating q i forward in time using Eq. (2).
- the invention utilizes a cooperative saddle straddle control strategy for a team of N ⁇ 3 robots to track the stable (and unstable) manifolds of a general conservative time-independent flow field F(x).
- the invention differs from the PIM procedure where it relies solely on information gathered via local sensing and shared through the network.
- a straight implementation of the PIM Triple Procedure necessitates global knowledge of the structure of the system dynamics throughout a given region given its reliance on computing escape times.
- the controller for the straddling robots consists of two discrete states: a passive control state, U P , and an active control state, U A .
- Robots execute U P until they reach the maximum allowable separation distance d Max from robot C.
- d Max the maximum allowable separation distance
- robots L and R are also constantly sampling the velocity of the local vector field and communicating these measurements and their relative positions to robot C. Robot C is then tasked to use these measurements to track the position of B S .
- w ij ⁇ circumflex over (x) ⁇ i (j) ⁇ q i ⁇ ⁇ 2 .
- bû B is included to ensure that the control strategy aims for a point in front of robot C rather than behind it.
- FIG. 2A shows the trajectories of three robots tracking a sinusoidal boundary while FIG. 2B shows the team tracking a 1D star-shaped boundary.
- the team maintains a saddle straddle formation across the boundary.
- the vector ⁇ is a 3 ⁇ 1 vector whose entries are given by [0,0, ⁇ (x,y)] T where ⁇ (x,y) is the curve describing the desired boundary.
- the estimated position of the boundary is given by the position of the tracking robot, i.e., robot C. In these examples, we filtered the boundary position using a simple first-order low pass filter.
- FIG. 3A shows the trajectories of the robots tracking a star shaped boundary.
- FIG. 3B is a snapshot of the experimental run.
- x . - ⁇ ⁇ ⁇ A ⁇ ⁇ sin ⁇ ( ⁇ ⁇ ⁇ f ⁇ ( x , t ) s ) ⁇ cos ⁇ ( ⁇ ⁇ ⁇ y s ) - ⁇ ⁇ ⁇ x + ⁇ 1 ⁇ ( t ) . ( 5 ⁇ a ) y .
- the trajectories of the straddling robots are shown in black and the estimated LCS is shown in white.
Abstract
Description
dx i /dt=V i cos θi +u i, (1a)
dy i /dt=V i sin θi +v i; (1b)
where xi=[xi,yi]T is the vehicle's planar position, Vi and θi are the vehicle's linear speed and heading, and ui=[ui,vi]T is the velocity of the fluid current experienced/measured by the ith vehicle. Additionally, we assume each agent can be circumscribed by a circle of radius r, i.e., each vehicle can be equivalently described as a disk of radius r.
dx/dt=F(x). (2)
In essence, ui=Fx(xi) and vi=Fy(xi). Let BS and BU denote the stable and unstable manifolds of Eq. (2). In general, BS and BU are the separating boundaries between regions in phase space with distinct dynamics. For 2D flows, B* are simply one-dimensional curves where * denotes either stable (S) or unstable (U) boundaries. For a small region centered about a point on B*, the system is unstable in one dimension. Finally, let ρ(B*) denote the radius of curvature of B* and assume that the minimum of the radius of curvature ρmin(B*)>r. This last assumption is needed to ensure the robots do not lose track of the B* due to sharp turns.
We note that while the primary control objective for robots L and R is to maintain a straddle formation across BS, robots L and R are also constantly sampling the velocity of the local vector field and communicating these measurements and their relative positions to robot C. Robot C is then tasked to use these measurements to track the position of BS.
where wij=∥{circumflex over (x)}i(j)−qi∥−2. Rather than rely solely on the current measurements provided by the three robots, it is possible to include the recent history of ûi(t) to improve the estimate of u(qk), i.e., ûi(t−ΔT), ûi(t−2ΔT), and so on, where ΔT is the sampling period and i={L,C,R}. Thus, the control strategy for the tracking robot C is given by
V C=∥[(q B +bû B)−x C ]−u C∥ (4a)
θC=βC (4b)
where βC denotes the difference in the heading of robot C and the vector (qB−ûB) and b>r is a small number. The term bûB is included to ensure that the control strategy aims for a point in front of robot C rather than behind it. As such, the projected saddle straddle line segment Ĵt at each time step is given by pc=qC+buC with Ĵt orthogonal to BS at qC and ∥Ĵt∥ chosen to be in the interval [2dMin2dMax].
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US9098079B2 (en) * | 2013-12-13 | 2015-08-04 | King Fahd University Of Petroleum And Minerals | Method of joint planning and control of a rigid, separable non-holonomic mobile robot using a harmonic potential field approach |
WO2016076923A1 (en) | 2014-11-14 | 2016-05-19 | Ocean Lab, Llc | Navigating drifter |
CN104635744B (en) * | 2014-12-18 | 2017-06-06 | 西北工业大学 | A kind of autonomous underwater carrier Random Coupling multi-load lays method |
CN107168309B (en) * | 2017-05-02 | 2020-02-14 | 哈尔滨工程大学 | Behavior-based multi-underwater robot path planning method |
CN107677272B (en) * | 2017-09-08 | 2020-11-10 | 哈尔滨工程大学 | AUV (autonomous Underwater vehicle) collaborative navigation method based on nonlinear information filtering |
CN110441736B (en) * | 2019-07-26 | 2021-05-07 | 浙江工业大学 | Multi-joint underwater unmanned vehicle variable baseline three-dimensional space positioning method |
CN111427358B (en) * | 2020-04-16 | 2021-07-13 | 武汉理工大学 | Navigation track control method and system for ship formation and storage medium |
CN112363517B (en) * | 2020-10-09 | 2021-09-28 | 中国科学院沈阳自动化研究所 | Formation control method for underwater glider |
CN113110458B (en) * | 2021-04-19 | 2023-09-01 | 大连海事大学 | Unmanned ship virtual target tracking control system |
CN113110467B (en) * | 2021-04-22 | 2024-02-02 | 大连海事大学 | Unmanned ship formation planning and guidance method and system under switching communication topology |
CN113268064B (en) * | 2021-07-01 | 2022-09-27 | 黄山学院 | Multi-mobile-robot cooperative formation control method considering communication time delay |
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